Multiphase and Interfacial flows have many applications in science and industries. In these flows, there are generally a free surface between different phases, where, there is surface tension that is physically important, and it is critical to find its value prescicely.Therefore, the main problem in these flows is to find the exact location of the interface between different phases and calculate the surface tension. In the field of direct numerical simulation, Front-Tracking method, uses a lagrangian grid which moves with the interface and the marker function is reconstructed by the location of the moving grid or front. By using this approach the numerical diffusion is removed and the location of each phase can be found with more precision. However, there are some drawbacks for this method which mainly refer to the complexities in using a separate computational grid for the interface and changing the topology of the front in case of coalescence and breakup. In this research, a new topology changing alghorithm is developed and used to simulate two physical examples in breakup of droplets in shear flow and droplets collision. There are three main advantages for this new alghorthm. Firstly, it changes the toplogy of the fronts globally which means doing all the operations at each step on all the elements of the all interfaces. Secondly, it can control the exact time of the breakup and coalescene by using a distance parameter as a criteria for starting the operations. Finally, it can handle both thin films that rupture or thin threads that snap. Deformation phenomenon and drops breakup in shear flow were studied using Front-Tracking method. At first, drop deformation in shear flow was studied in Stokes regime and was compared with experimental and numerical data. Break-up simulation was first carried out in Stokes regime. To validate the method performance and topology changing algorithm, results were compared with other numerical methods. The effect of the closeness parameter is analysed and it is seen that decreasing this parameter will delay the time of the topology changing. Moreover, breakup of a droplet in supercritical conditions is investigated to examine the ability of the method in simulating thin thread, satellite and subsatellite droplets and capillary instabilities. To simulate the coalescence phenomenon, a body force is used to move the droplets and the force is disappeared when droplets are close to each other. Again it is seen that by decreasing the closeness parameter the exact time of coalescence is increased. Moreover, by simulating the collision with different impact parameters and weber numbers all regimes of coalescence have been predicted. Comparing the results with the experiments in the literature, a good qualitative agreement was observed that shows the ability of the method and the present topology changing algorithm in simulating droplets coalescence. To sum up, the present method is able to simulate other multiphase flow problems that to some extent include collision and breakup of drops. Keywords: Multiphase Flows, Front-Tracking Method, Interface, Coalescence, Breakup, Topology changing alghorithm